Theorems · Theorem · several complex variables

ContinuousLinearMap.comp_hasFPowerSeriesWithinOnBall

∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [inst : NontriviallyNormedField 𝕜]
  [inst_1 : NormedAddCommGroup E] [inst_2 : NormedSpace 𝕜 E] [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F]
  [inst_5 : NormedAddCommGroup G] [inst_6 : NormedSpace 𝕜 G] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {s : Set E}
  {x : E} {r : ENNReal} (g : F →L[𝕜] G),
  HasFPowerSeriesWithinOnBall f p s x r → HasFPowerSeriesWithinOnBall (⇑g ∘ f) (g.compFormalMultilinearSeries p) s x r

If a function f has a power series p on a ball within a set and g is linear, then g ∘ f has the power series g ∘ p on the same ball.

Defined in

Mathlib.Analysis.Analytic.Basic

Cited by

5 results in Mathlib

Foundations

Depth 174 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace